An isoperimetric deficit upper bound of the convex domain in a surface of constant curvature

Abstract

In this paper, we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface X of constant curvature , that is, an isoperimetric deficit upper bound of the convex domain in X . The result is an analogue of the known Bottema's result of 1933 in the Euclidean plane E 2 . KeywordsKinematic formula, the surface of constant curvature, isoperimetric deficit, convex set MSC(2000): 52A10, 52A22, 52A55, 53C65 Citation: Li M, Zhou J Z. An isoperimetric deficit upper bound of the convex domain in a surface of constant curvature.

A sharp reverse Bonnesen-style inequality and generalization

“…where c is a constant andà is the area ofK , the domainK is bounded by the locus of the curvature centers of ∂K , where the equality sign holds if and only if K is a disc, that is,K is a point. Some reverse Bonnesen-style inequalities for surface X 2 of constant curvature have been obtained in [13,23,27,28]. Zhou et al obtained some reverse Bonnesen-style inequalities for any convex domain in [33].…”

On the Isoperimetric Deficit Upper Limit

“…Recently, the Bottema's inequality (5) has been generalized to the plane of constant curvature in [19].…”

“…An inequality of type (3) is called a Bonnesen style inequality. See [2], [3], [4], [6], [10], [16], [19], [21], [24], [25], [34] and [33] for more detailed references.…”

Some New Bonnesen-Style Inequalities

“…We recently investigate convex sets in a plane E 2 ϵ of constant curvature ϵ and obtain an upper limit for the isoperimetric deficit of a convex domain K in E 2 ϵ with continues radius ρ ϵ of ∂K (see [19]). The upper limit can be seen as a generalization of the Bottema's result in E 2 ϵ and involves the smallest radius ρ m and the greatest radius ρ M of curvature radius ρ ϵ of ∂K.…”